Question

The extension due to self-weight of a bar of uniform cross section being hanged vertically downward is ___ times the extension produced by the same weight applied at the lower end of the vertical bar.

• A. 0.5
• B. 20
• C. 0.333
• D. 0.667

Hint:

When a weight is applied to a vertically hanging bar, its own weight causes a smaller extension compared to the same weight applied at the lower end. The self-weight causes approximately half the extension.

Answer 1

The Correct Option is A

Step 1: Extension Due to Weight Applied at the Lower End When a weight is applied at the lower end of a vertically hanging bar, the extension is given by: \[ \Delta L_{\text{end}} = \frac{F L}{A Y} \] where \( F \) is the force, \( L \) is the length of the bar, \( A \) is the cross-sectional area, and \( Y \) is the Young's Modulus.
Step 2: Extension Due to Self-Weight of the Bar When the same weight acts due to the self-weight of the bar, the extension is calculated by integrating the elongation over the length of the bar: \[ \Delta L_{\text{self}} = \frac{1}{2} \times \frac{F L}{A Y} \] This means the extension due to the self-weight is half the extension when the same weight is applied at the lower end.
Step 3: Ratio of Extensions Thus, the ratio of the extensions is: \[ \frac{\Delta L_{\text{self}}}{\Delta L_{\text{end}}} = \frac{1}{2} \]
Step 4: Conclusion The extension due to self-weight is 0.5 times the extension produced by the same weight applied at the lower end of the bar.